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A New Tool for Nonlinear Dynamical Analysis of Heart Rate Variability A. Posiewnik and J. Dąbkowski Institute of Theoretical Physics and Astrophysics, University of Gdansk, 80-952 Gdansk, Wita Stwosza 57, Poland

Z. Naturforsch. 53a, 112-116 (1998); received November 29, 1997

In this paper we analyse the sequences of the time intervals between heart-beats-the RR intervals- by means of AIP (artificial insymmetration patterns) diagrams.

The sequences were produced by artificial heartbeat sequences generated numerically and compared with sequences obtained from real heart activity.

We hope that the AIP diagrams method will prove useful for a rapid qualitative assessment of dynam-ics from nonlinear time series, and that it is able to distinguish various types of heart dynamics (regu-lar and pathological), while other diagnostical methods fail.

In industrial countries, sudden cardiac death (SCD) is a frequent cause of death among men aged 20-60 years. So it is important to recognize patients with high risk of SCD as early and reliably as possible.

The analysis of electrocardiogram (ECG) data may be divided into two basic areas. In morphological analysis the shape of electrical pulses measured is examined [1]. The method is based on the computation of the respec-tive Shannon entropies and often provides a sharp dis-tinction between healthy persons and patients with high risk of SCD. On the other hand, in the complementary type of ECG analysis the time distances between heart-beats - the RR intervals - are measured [2].

Three-dimensional images in the phase space are formed by means of the Takens-Ruelle reconstruction method of long sequences of cardiac interbeat intervals [3]. Projections of the three-dimensional images of the RR sequences could serve as a diagnostic tool of differ-ent cases of cardiac arrhythms [3, 4],

In this paper we used the algorithm for so called arti-ficial insymmetration patterns in order to analyse RR se-quences.

Artificial insymmetration patterns (AIP) (also known as symmetrized dot patterns), have been introduced by Pickover [5] as a qualitative method of visualizing cor-relation functions in time series data. For the interpreta-tion of underlying patterns, the data are mapped in a man-ner which artifically induces symmetry into the data set.

This artificial symmetry is statistically enhanced for portions of the time series signal that have a high prob-ability of contributing to the underlying signal, while

Reprint requests to A. Posiewnik; e-mail: fizap@paula.univ.gda.pl; Fax: 485 8341 31 75.

those data points dominated by noise produce a scattered field with no underlying symmetry. The algorithm is im-plemented by working with only a single channel of data. We process the data by selecting sequential points in the data set to be the radius and angle coordinates in polar space.

The first value is the radius while the second is used as the angle. The angle point is also duplicated by a sec-ond point with an opposing sign value. The resulting po-lar spatial distribution is reproduced around the polar cen-ter for an arbitrary number of times, by specifying the number of segments the 360 degree circle is to be divid-ed into. 60 degrees seems to be a good working number, although in practice 120 degrees was found to enhance certain data structures.

The AIP transformation is given by

X:F(t)->S(rj, 0ip <2>y)

where F (t) represents the discrete time series, and S is the AIP. S is a traditional function of r in radial vector polar coordinates, while 0 and <f> are two polar angles obtained by Pickovers's transformation:

; H-l *

Q Q'lhlZl^ 1 H-L

1 H-L

where j=\,2,3,...,N- 1.

0' = (3607m) i, i = 1,2, 3 m.

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114 A. Posiewnik and J. D^bkowski • A New Tool for Nonlinear Dynamical Analysis of Heart Rate Variability 113

Here N is the number of points in the time series, m is the number of symmetric mirrored or conjugate plane reflec-tions, H the maximum value in the data set, L the mini-mum value in the data set, and «^represents the maximum value used to normalize or scale the data. Pickover sug-gests that > 360Im, which was found to be a good work-ing figure of merit.

The resulting AIP image can be used to visually dis-criminate data sets that appear very similar and virtual-ly indistinguishable in a time series. The actual symme-tric dot patterns by themselves are of limited use since they do not quantify the underlying data but rather only provide a qualitative insight. Jaenisch has suggested that such patterns may lend themselves to processing and rec-ognition by neural networks [6].

He also proposed to characterize AIP patterns in terms of correlation functions and fractal dimensions using op-tical parallel processing. Such techniques would enable real time processing using collected data mapped into an AIP.

We investigated the characterization of AIP patterns by means of histograms - and results were very encour-aging.

We used a simple algorithm for which the Fortran 77 code is provided by Pickover*. In practice, good AIP pat-terns require approximately 200 points to produce reli-able characteristics; however, smaller data sets were al-so used, instead of sufficient initial data, and the results were encouraging.

We checked our method by applying it to numerical-ly created data and comparing them with digitized elec-trocardiograms obtained from patients with different pathological heart rhythms.

AIP diagrams can assess subtle correlations which may exist in data sets and are not easily detectable with oth-er techniques. The AIP algorithm is very simple and com-putationally inexpensive, and it can also be applied to short data sets which could be purely deterministic, sto-chastic or mixed, and not necessarily stationary. The method has a very appealing "visual" aspect. It is our hope that the AIP diagrams could distinguish between various cases of pathology and health, often also in cases where other methods fail.

AIP diagrams obtained from artificially generated heart beat sequences can be used as a catalogue since by its construction the nature of each pattern is well defined.

* The Fortran 77 code for AIP algorithm is available upon request; contact A. Posiewnik (e-mail: fizap@paula.univ. gda.pl.)

AIP diagrams obtained from real ECG series could thus be compared with this catalogue and a possible corre-sponding disease be identified.

In our analysis we used artificial heart beat sequences generated according to the prescriptions given in the Grossmann-Ranft paper [7]. There are four main classes of sequences, and we now give a short description of each.

1. Sinus Rhythm

All RR intervals xn,n= 1, 2 , . . . are equal but are pol-luted by noise of physiological origin. So RR intervals are presented by the map

xn+i =xn-c • (xn-\) + a- gn,

xne [0,2],/i = 0, 1,2, ...cca. Here is normalized gaussian white noise (£„) = 0,

(Ci, O = Snm a n d o is its strenght. The restoring term c (xn— 1) prevents the signal sequence from diffusional broadening.

This model [7] generates a sort of cudgel in the phase space diagram, Figure 1. We generated the AIP diagrams for the model, choosing different values of c, dand com-pared them with AIP diagrams obtained from ECG's of healthy persons. It is easy to see the structural similarity of all these diagrams, Fig. 2a, 2b.

2. Extrasystoles with Fixed Coupling

Extrasystoles are extra beats of the heart. Unlike for the sinus rhythm, their electrical origin is not the sinus node. Grossmann and Ranft [7] used the following algo-rithm for generating heart beat sequences with extrasys-toles: • If the last heart beat was an extrasystole, then a com-pensatory pause will follow. • If the last heart beat was not an extrasystole, then the next one will again be a sinus beat with probability p and an extrasystole with probability 1 -p.

The generated RR sequence typically reads: " . . . x x y 2 x-y xy 2x-y xx. . . " wherex is the distance between two successive sinus beats, y is the time lag between an extrasystole and the preceding beat (ectopic beat), denot-ed as coupling interval, 2 x-y is the lenght of the com-pensatory pause to follow, y = y0 is constant (fixed coupling interval). The sinus beats are again modelled by the sequence xn from class 1.

In the phase space diagrams one now obtains four cud-gels in the projection on the (xn, xn+l) coordinate plane, Figure 3. The corresponding AIP diagrams created by ar-

114 A. Posiewnik and J. D^bkowski • A New Tool for Nonlinear Dynamical Analysis of Heart Rate Variability 114

tificial sequences are presented in Fig. 4a and the AIP di-agrams obtained from E-CG's of long-time survivors are presented in Figure 4b. As in class 1, one can easily see the apparent structural similarity.

3. Extrasystoles with Variable Coupling

• One can choose a constant sinus rhythm x0 and ad-mit variable coupling intervals. The successive coupling intervals are then modelled by the same algorithm as for the sinus rhythm in class 1.

• Alternatively, both x and y are taken as variable, i.e. x and y are generated by the noise polluted model from

class 1. For the frequency of the extrasystole occurrence we used the (p, 1 -p) algorithm.

A typical phase-space diagram is given in Figure 5. The corresponding AIP diagrams for artificially generat-ed sequences and for severely ill patients are displayed in Figs. 6a and 6b.

4. Conclusions

1. From our analysis we conclude, similarily to Gross-mann and Ranft, that one can represent different pathol-ogies in heart beat sequences by simple models with noise polluted regular beats eventually with occasional extra-

Ä l f c IF

• ••; WA y

i Ä V? - M M

i f

a

Fig. 2a. AIP diagram of class 1 generated by artificial data using c = 0.006, a = 0.02. Fig. 2b. AIP diagram of real ECG data from healthy person.

114 A. Posiewnik and J. D^bkowski • A New Tool for Nonlinear Dynamical Analysis of Heart Rate Variability 115

X

Fig. 3. Typical phase space diagram of class 1 and additional extrasystoles de-fined in class 2.

Fig. 4a. AIP diagram with artificial heart beat sequence as defined in class 2, c = 0.006, <7 = 0.02. Fig. 4b. AIP diagram of real ECG data from a patient with ventricular extrasystoles.

systoles, where regular or irregular occurrence is re-flected in the complexity of the structures in the phase space.

2. The AIP diagrams can serve as reliable diagnostic tools for recognizing different disturbances in the hearth rhythm.

Now we are analysing the finer details of the AIP di-agrams in order to more precisely distinguish the vari-ous disturbances in the heart rhythm. It is very promis-ing to use color in generating the AIP diagrams.

The authors acknowledge the partial support of BW/5400-5-0303-7.

116 A. Posiewnik and J. D^bkowski • A New Tool for Nonlinear Dynamical Analysis of Heart Rate Variability

2.0

1.5 -

T 1.0 c X

0.5 -

0.0 0.0

• °<%

o & ° ; • • » /

* • « « / u f .

0.5 1.0

Xn

1.5 2.0

Fig. 5. A phase space diagram of mod-el 1 with variable regular beat sequence and variable extrasystole as defined in class 3, c = 0.006, <7=0.02.

b

Fig. 6a. AIP diagram corresponding to the phase space diagram of previous figure. Fig. 6b. AIP diagram of real ECG data from severely ill patient.

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[5] C. A. Pickover, Computers, Pattern, Chaos, and Beauty, (St. Martin's Press, New York 1990.

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